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In baseball, is there a linear correlation between batting average and home run percentage? Let \(x\) represent the batting average of a professional baseball player, and let \(y\) represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of \(n=7\) professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan Publishing Company). (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Use a calculator to verify that \(\Sigma x=1.957, \Sigma x^{2} \approx 0.553, \Sigma y=30.1\) \(\Sigma y^{2}=150.15,\) and \(\Sigma x y \approx 8.753 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Step by step solution

01

Construct the Scatter Plot

To create the scatter plot, plot each player's batting average as the x-coordinate and their home run percentage as the y-coordinate on a graph. After plotting all points, draw the line that seems to best fit through the data points. This line represents the trend between the batting average and home run percentage.

02

Analyze the Scatter Plot

Observe the points on the scatter plot to determine the nature of the correlation. If the points are close to a line and slope upward, the correlation is positive and could be moderate or high. If they slope downward, the correlation is negative.

03

Calculate the Correlation Coefficient

Use the given sums to calculate the correlation coefficient, \( r \), using the formula: \[ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{(n \Sigma x^2 - (\Sigma x)^2)(n \Sigma y^2 - (\Sigma y)^2)}} \]. Substitute the given values: \( \Sigma x = 1.957, \Sigma x^2 = 0.553, \Sigma y = 30.1, \Sigma y^2 = 150.15, \Sigma xy = 8.753 \), and \( n = 7 \).

04

Substitute and Compute

First, calculate the numerator: \(7 \times 8.753 - 1.957 \times 30.1 = 61.271 - 58.9057 = 2.3653 \). Next, compute the denominator: \( \sqrt{(7 \times 0.553 - 1.957^2)(7 \times 150.15 - 30.1^2)} \). Calculate each component: \(7 \times 0.553 = 3.871 \), \(1.957^2 = 3.832049 \), \(7 \times 150.15 = 1051.05 \), \(30.1^2 = 906.01 \). Thus, \( \sqrt{(3.871 - 3.832049)(1051.05 - 906.01)} = \sqrt{0.038951 \times 145.04} = \sqrt{5.64146204} \approx 2.37523\), leading to \(r = \frac{2.3653}{2.37523} \approx 0.996 \).

05

Interpret the Correlation Coefficient

A correlation coefficient \( r \) of \( 0.996 \) indicates a high positive correlation between batting average and home run percentage. This means that as the batting average \( x \) increases, home run percentage \( y \) tends to increase as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is a crucial visual tool that helps us understand the relationship between two variables. In this case, plotting the batting average as the x-axis and the home run percentage as the y-axis gives us a clear picture. Each point on the scatter plot represents a single player's batting average and home run percentage.
To begin, you place the values of the independent variable (here, batting average) along the horizontal axis. Next, place the dependent variable (home run percentage) along the vertical axis. Each data pair (x, y) is plotted. After all points are placed on the graph, it may be helpful to draw a line of best fit.
The line of best fit is not merely a visual aid; it helps in visualizing the correlation type, whether positive, negative, or neutral.

Correlation Coefficient

The correlation coefficient, often represented as the letter \( r \), quantifies the degree to which two variables are related. It ranges from -1 to 1.
An \( r \) value close to 1 suggests a strong positive relationship, indicating that as one variable increases, the other tends to increase as well. An \( r \) value near -1 indicates a strong negative relationship, meaning as one variable increases, the other decreases. If \( r \) is near zero, it suggests little to no linear correlation, meaning the variables don't have a consistent linear relationship.
In this exercise, the formula to calculate \( r \) is: \[ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{(n \Sigma x^2 - (\Sigma x)^2)(n \Sigma y^2 - (\Sigma y)^2)}} \] Substituting the provided values allows for precise calculation and interpretation of the data.

Linear Correlation

Linear correlation refers to a straight-line relationship between two variables. In our example, linear correlation helps determine how consistently a player's batting average is related to their home run percentage.
When examining linear correlation, the resulting line from the scatter plot plays a pivotal role. If the line slopes upwards as you move from left to right, the correlation is positive. Conversely, if it slopes downwards, the correlation is negative.
Furthermore, linear correlation is often specified in terms of the strength of the relationship. This category includes terms such as strong, moderate, or weak correlation to indicate how closely the data points are clustered around the line of best fit.

Positive Correlation

A positive correlation signifies that both variables move in the same direction. In the context of baseball performance, a positive correlation between batting average and home run percentage means that players with higher batting averages also tend to have higher home run percentages.
This positive relationship indicates an alignment between hitting consistency and power-hitting capability. Observing this trend can be significant for trainers and players who aim to enhance performance metrics.
A graph showing positive correlation will have points that trend upward from left to right. It's a visual representation that as one player's batting average increases, their home run percentage is likely increasing too. This type of correlation supports the idea that certain skills, such as hitting accuracy, are linked with power hitting.